Spatial Bears - Elevation
# Loads data
load("bc_bear_occurrences.Rda")
load("BC_Covariates.Rda")
summary(DATA)## Length Class Mode
## Window 1 SpatialPolygons S4
## Elevation 10 im list
## Forest 10 im list
## HFI 10 im list
## Dist_Water 10 im listVisualize bear occurrences in BC
# Extracts location columns
bears_loc <- occ_data[, c("decimalLongitude", "decimalLatitude", "month", "year")]
bears_loc_filtered <- subset(bears_loc, year %in% c(2020, 2021, 2022, 2023, 2024))
# Creates sf object
bears_sf <- st_as_sf(
bears_loc_filtered,
coords = c("decimalLongitude", "decimalLatitude"),
crs = 4326 # WGS84 (longitude/latitude)
)
# BC Albers projection
bears_sf_proj <- st_transform(bears_sf, crs = 3005)
# Extracts coordinates
coords <- st_coordinates(bears_sf_proj)
# Extracts BC window
window_sf <- st_as_sf(DATA$Window)
window_proj <- st_transform(window_sf, crs = 3005)
window <- as.owin(window_proj)
# Creates ppp object
bears_ppp <- ppp(
x = coords[,1],
y = coords[,2],
window = window
)## Warning: 187 points were rejected as lying outside the specified window## Warning: data contain duplicated pointsplot(bears_ppp, pch = 21, main = "Black bear occurrences in BC, 2024")## Warning in plot.ppp(bears_ppp, pch = 21, main = "Black bear occurrences in BC,
## 2024"): 187 illegal points also plotted
Compare seasonal distributions
# Defines the seasons and corresponding colors
seasons <- list(
winter = c(12, 1, 2),
spring = c(3, 4, 5),
summer = c(6, 7, 8),
autumn = c(9, 10, 11)
)
# Create an empty list to store the ppp objects for each season
ppp_list <- list()
# Create an empty list to store the filtered data for each season
bears_sf_list <- list()
# Loop over seasons
for (i in 1:length(seasons)) {
# Filter for each season
season_name <- names(seasons)[i]
season_months <- seasons[[i]]
# Filter bears_loc_filtered by the season's months
bears_loc_season <- bears_loc_filtered[bears_loc_filtered$month %in% season_months, ]
# Print the number of observations for this season
cat(season_name, ": ", nrow(bears_loc_season), " observations\n", sep="")
# Create sf object for the filtered season
bears_sf_season <- st_as_sf(
bears_loc_season,
coords = c("decimalLongitude", "decimalLatitude"),
crs = 4326 # WGS84 (longitude/latitude)
)
# Transform to BC Albers projection
bears_sf_season_proj <- st_transform(bears_sf_season, crs = 3005)
# Store sf object in the list
bears_sf_list[[season_name]] <- bears_sf_season_proj
# Extract x, y coordinates
coords <- st_coordinates(bears_sf_season_proj)
# Create ppp object for each season
suppressWarnings(
bears_ppp_season <- ppp(
x = coords[, 1],
y = coords[, 2],
window = window
)
)
# Store ppp object in the list
ppp_list[[season_name]] <- bears_ppp_season
}## winter: 76 observations
## spring: 808 observations
## summer: 1862 observations
## autumn: 841 observations# Plot the four maps in a 2x2 grid
par(mfrow=c(2,2), mar=c(1,1,1,1)) # Set up a 2x2 plotting window
# Loop over ppp objects and plot them
for (i in 1:length(ppp_list)) {
season_name <- names(ppp_list)[i]
suppressWarnings(
plot(ppp_list[[season_name]],
main = paste(season_name, "(2024)"),
pch = 21)
)
}
plot(DATA$Elevation)
plot(DATA$Forest)
plot(DATA$HFI)
plot(DATA$Dist_Water)
elev <- DATA$Elevation
cover <- DATA$Forest
dist_water <- DATA$Dist_Water
hfi <- DATA$HFIEstimating the intensity of bear occurrences based on elevation
# intensity as a function of elev
par(mfrow=c(2,2), mar=c(2,2,2,2))
for (i in 1:length(ppp_list)) {
season_name <- names(ppp_list)[i]
rho <- rhohat(ppp_list[[season_name]], elev)
plot(rho, xlim=c(0, max(elev)), main = paste(season_name))
}
# Loads elevation raster
elev <- DATA$Elevation
# Converts im to raster
im2raster <- function(im_obj) {
m <- as.matrix(im_obj)
r <- raster(m)
ext <- c(im_obj$xrange[1] - im_obj$xstep/2,
im_obj$xrange[2] + im_obj$xstep/2,
im_obj$yrange[1] - im_obj$ystep/2,
im_obj$yrange[2] + im_obj$ystep/2)
extent(r) <- ext
return(r)
}
elev_raster <- im2raster(elev)
# Extracts elevation values at occurrence locations
bears_sf_proj$elev_value <- extract(elev_raster, bears_sf_proj)
summary(bears_sf_proj$elev_value)## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## 11.68 461.09 644.88 754.84 937.91 2308.18 1499KDE to estimate the concentration of bear occurrences based on elevation values. The elevation data for bear locations ranges from a minimum of 11.68 meters to a maximum of 2308.18 meters, with most occurrences concentrated between 461.09 meters and 937.91 meters. The mean elevation is 754.84 meters, slightly higher than the median, indicating a skew towards higher elevations.
# KDE for Bear Locations
kde_b <- density(bears_sf_proj$elev_value, na.rm = TRUE, bw = "SJ-dpi")
summary(bears_sf_proj$elev_value)## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## 11.68 461.09 644.88 754.84 937.91 2308.18 1499plot(kde_b,
main = "Kernel Density Estimate of Elevation at Bear Locations",
xlab = "Elevation (meters)",
ylab = "Density",
col = "#2C6B2F",
lwd = 2) 
Fitted model: Intercept: -17.37; Elevation coefficient: -0.0025. The negative coefficient for elevation indicates that, as elevation increases, the expected intensity of bear occurrences decreases. This suggests that bears are less likely to be found at higher elevations.
# NA with mean elevation
mean_elev <- mean(as.vector(as.matrix(elev)), na.rm = TRUE)
elev_clean <- eval.im(ifelse(is.na(elev), mean_elev, elev))
# model
elevation_model <- ppm(bears_ppp, ~ elev, covariates = list(elev = elev_clean))
print(elevation_model)## Nonstationary Poisson process
## Fitted to point pattern dataset 'bears_ppp'
##
## Log intensity: ~elev
##
## Fitted trend coefficients:
## (Intercept) elev
## -17.36871576 -0.00253243
##
## Estimate S.E. CI95.lo CI95.hi Ztest
## (Intercept) -17.36871576 2.963479e-02 -17.426798885 -17.310632643 ***
## elev -0.00253243 4.234993e-05 -0.002615434 -0.002449426 ***
## Zval
## (Intercept) -586.09208
## elev -59.79774# Predicts and plots intensity
pred_intensity_elev <- predict(elevation_model)
plot(pred_intensity_elev,
main = "Predicted Intensity Based on Elevation")
points(bears_ppp,
pch = 20,
cex = 0.4,
col = "#B24A2F")
Computing and plotting residuals of the point process model fitted to elevation data. Using Pearson residuals and removing any non-finite residuals
elevation_residuals <- residuals(elevation_model, type = "pearson")
elevation_residuals$v[!is.finite(elevation_residuals$v)] <- NA
plot(elevation_residuals,
main = "Residuals - Elevation Model",
na.col = "transparent")
Winter: Intercept: -20.28; Elevation Coefficient: -0.0049. For winter, the negative coefficient for elevation indicates that bear occurrences decrease as elevation increases. The model did not converge, so caution is needed in interpreting the results.
Spring: Intercept: -18.94; Elevation Coefficient: -0.0024. In spring, the relationship between elevation and bear occurrences is also negative, with a slight decrease in occurrences as elevation increases. However, the model showed some issues with convergence.
Summer: Intercept: -18.21; Elevation Coefficient: -0.0022. Summer shows a similar negative relationship, with bear occurrences decreasing slightly as elevation increases.
Autumn: Intercept: -18.51; Elevation Coefficient: -0.0031. In autumn, bear occurrences also decrease with elevation, with a slightly stronger negative coefficient compared to spring and summer.
####All seasons show a negative relationship between bear occurrences and elevation. Winter has the largest negative coefficient (-0.0049), meaning a stronger decrease in occurrences with elevation.
# Seasonal elevation models
models_elev_seasonal <- list()
for (season in names(ppp_list)) {
models_elev_seasonal[[season]] <- ppm(ppp_list[[season]], ~ elev, covariates = list(elev = elev_clean))
cat("Model for", season, ":\n")
print(models_elev_seasonal[[season]])
}## Warning: glm.fit: algorithm did not converge## Model for winter :
## Nonstationary Poisson process
## Fitted to point pattern dataset 'ppp_list[[season]]'
##
## Log intensity: ~elev
##
## Fitted trend coefficients:
## (Intercept) elev
## -20.281500393 -0.004896402
##
## Estimate S.E. CI95.lo CI95.hi Ztest
## (Intercept) -20.281500393 0.1590970890 -20.593324958 -19.96967583 ***
## elev -0.004896402 0.0004162485 -0.005712234 -0.00408057 ***
## Zval
## (Intercept) -127.47876
## elev -11.76317
## *** Fitting algorithm for 'glm' did not converge ***
## Model for spring :
## Nonstationary Poisson process
## Fitted to point pattern dataset 'ppp_list[[season]]'
##
## Log intensity: ~elev
##
## Fitted trend coefficients:
## (Intercept) elev
## -18.938423756 -0.002390713
##
## Estimate S.E. CI95.lo CI95.hi Ztest
## (Intercept) -18.938423756 6.296883e-02 -19.061840399 -18.815007114 ***
## elev -0.002390713 8.708531e-05 -0.002561397 -0.002220029 ***
## Zval
## (Intercept) -300.75870
## elev -27.45254
## Model for summer :
## Nonstationary Poisson process
## Fitted to point pattern dataset 'ppp_list[[season]]'
##
## Log intensity: ~elev
##
## Fitted trend coefficients:
## (Intercept) elev
## -18.213839873 -0.002227723
##
## Estimate S.E. CI95.lo CI95.hi Ztest
## (Intercept) -18.213839873 4.265390e-02 -18.297439975 -18.13023977 ***
## elev -0.002227723 5.699771e-05 -0.002339437 -0.00211601 ***
## Zval
## (Intercept) -427.01468
## elev -39.08443
## Model for autumn :
## Nonstationary Poisson process
## Fitted to point pattern dataset 'ppp_list[[season]]'
##
## Log intensity: ~elev
##
## Fitted trend coefficients:
## (Intercept) elev
## -18.514156648 -0.003139775
##
## Estimate S.E. CI95.lo CI95.hi Ztest
## (Intercept) -18.514156648 5.671064e-02 -18.625307457 -18.403005838 ***
## elev -0.003139775 9.455484e-05 -0.003325099 -0.002954451 ***
## Zval
## (Intercept) -326.46708
## elev -33.20586# Combines seasonal data
bears_all_elevation <- do.call(rbind, lapply(names(bears_sf_list), function(season) {
sf_season <- bears_sf_list[[season]]
coords <- st_coordinates(sf_season)
elev_vals <- extract(elev_raster, sf_season)
data.frame(x = coords[, 1], y = coords[, 2],
elev_value = elev_vals, season = season)
}))
bears_all_elevation$season <- as.factor(bears_all_elevation$season)
bears_all_elevation$count <- 1GAM results suggest that neither elevation nor season has a significant influence on the spatial distribution of bear locations. Warnings about the model fitting process suggest that the results may not be reliable.
# GAM with elevation and season
gam_elevation <- gam(count ~ s(elev_value) + season, data = bears_all_elevation, family = poisson())
summary(gam_elevation)##
## Family: poisson
## Link function: log
##
## Formula:
## count ~ s(elev_value) + season
##
## Parametric coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 6.729e-17 4.510e-02 0 1
## seasonspring 2.172e-22 6.531e-02 0 1
## seasonsummer 7.375e-22 5.431e-02 0 1
## seasonwinter 8.153e-21 1.544e-01 0 1
##
## Approximate significance of smooth terms:
## edf Ref.df Chi.sq p-value
## s(elev_value) 1 1.001 0 1
##
## R-sq.(adj) = NaN Deviance explained = NaN%
## UBRE = -0.99521 Scale est. = 1 n = 2088Model shows a negative relationship between bear locations and elevation, suggesting bears are less likely to be found at higher elevations. Quadratic term suggests a slight non-linear effect, but primary relationship appears negative. Model also indicates that summer sees the highest concentration of bear points, while winter has an expected decrease in bear points. Spring also shows a negative effect, although it is smaller than winter’s effect.
Bears tend to prefer lower elevations (based on the negative relationship with elevation). The summer season has the highest bear presence, while winter and spring have much lower bear occurrences, as to be expected.
# Polynomial ppm model with season
bears_all_elevation <- ppp(
x = bears_all_elevation$x,
y = bears_all_elevation$y,
window = window,
marks = bears_all_elevation$season
)## Warning: 187 points were rejected as lying outside the specified window## Warning: data contain duplicated pointsmarks(bears_all_elevation) <- data.frame(season = marks(bears_all_elevation))
model_poly_elevation <- ppm(bears_all_elevation, ~ polynom(elev, 2) + marks,
covariates = list(elev = elev_clean))
print(model_poly_elevation)## Nonstationary multitype Poisson process
## Fitted to point pattern dataset 'bears_all_elevation'
##
## Possible marks: 'autumn', 'spring', 'summer' and 'winter'
##
## Log intensity: ~elev + I(elev^2) + marks
##
## Fitted trend coefficients:
## (Intercept) elev I(elev^2) marksspring markssummer
## -1.850394e+01 -3.937116e-03 9.510112e-07 -3.957121e-02 7.891398e-01
## markswinter
## -2.379296e+00
##
## Estimate S.E. CI95.lo CI95.hi Ztest
## (Intercept) -1.850394e+01 4.652449e-02 -1.859513e+01 -1.841276e+01 ***
## elev -3.937116e-03 1.034097e-04 -4.139795e-03 -3.734437e-03 ***
## I(elev^2) 9.510112e-07 6.143974e-08 8.305915e-07 1.071431e-06 ***
## marksspring -3.957121e-02 5.053363e-02 -1.386153e-01 5.947288e-02
## markssummer 7.891398e-01 4.266227e-02 7.055233e-01 8.727563e-01 ***
## markswinter -2.379296e+00 1.215116e-01 -2.617454e+00 -2.141137e+00 ***
## Zval
## (Intercept) -397.7247657
## elev -38.0729907
## I(elev^2) 15.4787625
## marksspring -0.7830669
## markssummer 18.4973701
## markswinter -19.5808065# Compares models using AIC
cat("AIC for poly model: ", AIC(model_poly_elevation), "\n")## AIC for poly model: 142038.1cat("AIC for gam model: ", AIC(gam_elevation), "\n")## AIC for gam model: 4186.001Creating a point pattern object for each season
# Creates seasonal ppp objects
ppp_list <- list()
for (s in names(seasons)) {
bears_loc_season <- bears_loc_filtered[bears_loc_filtered$month %in% seasons[[s]], ]
bears_sf_season <- st_as_sf(bears_loc_season, coords = c("decimalLongitude", "decimalLatitude"), crs = 4326)
bears_sf_season_proj <- st_transform(bears_sf_season, crs = 3005)
coords_season <- st_coordinates(bears_sf_season_proj)
suppressWarnings(
ppp_list[[s]] <- ppp(x = coords_season[,1], y = coords_season[,2], window = window)
)
}
# Extracts covariates
elevation <- DATA$Elevation
cover <- DATA$Forest
hfi <- DATA$HFI
dist_water <- DATA$Dist_WaterCalculating and plotting the KDE for bear occurrences for each season
# KDE Surface Comparison
dens_list <- lapply(ppp_list, function(p) density(p, sigma = 10000))
for (s in names(dens_list)) {
plot(dens_list[[s]], main = paste("KDE Surface -", s))
}
Performing Quadrat test for each season to analyze the spatial pattern of bear occurrences. Dividing area into a grid of 5x5 and testing for deviations from a random distribution. Bear locations do not follow a random distribution and show spatial clustering across all seasons, meaning their presence is likely influenced by specific environmental or ecological factors that are season-dependent. The quadrant test results suggest that for all seasons (winter, spring, summer, and autumn), bear locations exhibit significant spatial clustering or non-random patterns rather than being randomly distributed (as indicated by the very small p-values).
Bear distribution is influenced by factors other than randomness, such as environmental features, resource availability, or behavior patterns that vary across seasons.
Winter: X2 = 561.39, p-value < 2.2e-16. P-value indicates a strong deviation from CSR for winter, suggesting that bear locations are not randomly distributed in the winter season.
Spring: X2 = 2202.7, p-value < 2.2e-16. Test statistic is very large, and the p-value is highly significant, indicating that the bear locations in spring also show non-random clustering or patterns.
Summer: X2 = 3767, p-value < 2.2e-16. Summer further supports a non-random pattern, with an even higher test statistic and significant p-value.
Autumn: X2 = 2720.5, p-value < 2.2e-16. Autumn shows a significant departure from CSR, confirming that bear locations are non-randomly distributed.
# quadrat tests
for (s in names(ppp_list)) {
qt <- quadrat.test(ppp_list[[s]], nx = 5, ny = 5)
print(qt)
plot(qt, main = paste("Quadrat test -", s))
}## Warning: Some expected counts are small; chi^2 approximation may be inaccurate##
## Chi-squared test of CSR using quadrat counts
##
## data: ppp_list[[s]]
## X2 = 561.39, df = 20, p-value < 2.2e-16
## alternative hypothesis: two.sided
##
## Quadrats: 21 tiles (irregular windows)
## Warning: Some expected counts are small; chi^2 approximation may be inaccurate##
## Chi-squared test of CSR using quadrat counts
##
## data: ppp_list[[s]]
## X2 = 2202.7, df = 20, p-value < 2.2e-16
## alternative hypothesis: two.sided
##
## Quadrats: 21 tiles (irregular windows)
## Warning: Some expected counts are small; chi^2 approximation may be inaccurate##
## Chi-squared test of CSR using quadrat counts
##
## data: ppp_list[[s]]
## X2 = 3767, df = 20, p-value < 2.2e-16
## alternative hypothesis: two.sided
##
## Quadrats: 21 tiles (irregular windows)
## Warning: Some expected counts are small; chi^2 approximation may be inaccurate##
## Chi-squared test of CSR using quadrat counts
##
## data: ppp_list[[s]]
## X2 = 2720.5, df = 20, p-value < 2.2e-16
## alternative hypothesis: two.sided
##
## Quadrats: 21 tiles (irregular windows)
Fitting an inhomogeneous Poisson ppm for each season, using elevation. Model used to predict intensity and plotted for each season.
# inhomogeneous poission model
for (s in names(ppp_list)) {
model <- ppm(ppp_list[[s]] ~ elev, covariates = list(elev = elevation))
plot(predict(model), main = paste("Intensity ~ Elevation -", s))
}## Warning: glm.fit: algorithm did not converge
## Warning: Values of the covariate 'elev' were NA or undefined at 0.02% (1 out of
## 5456) of the quadrature points. Occurred while executing: ppm.ppp(Q =
## ppp_list[[s]], trend = ~elev, data = NULL, interaction = NULL,
Fitting a multivariate Poisson Point Process model to bear occurrences for each season, considering all covariates: elevation, forest cover, hfi, and distance to water. Predicted intensity plotted for each season.
# multivariate ppm
for (s in names(ppp_list)) {
model <- ppm(ppp_list[[s]] ~ elev + cover + hfi + dist_water,
covariates = list(elev = elevation, cover = cover, hfi = hfi, dist_water = dist_water))
print(summary(model))
plot(predict(model), main = paste("Predicted Intensity -", s))
}## Warning: Values of the covariate 'hfi' were NA or undefined at 0.68% (4 out of
## 587) of the quadrature points. Occurred while executing: ppm.ppp(Q =
## ppp_list[[s]], trend = ~elev + cover + hfi + dist_water,## Warning: glm.fit: algorithm did not converge## Point process model
## Fitted to data: ppp_list[[s]]
## Fitting method: maximum likelihood (Berman-Turner approximation)
## Model was fitted using glm()
## *** Algorithm did not converge ***
## Call:
## ppm.formula(Q = ppp_list[[s]] ~ elev + cover + hfi + dist_water,
## covariates = list(elev = elevation, cover = cover, hfi = hfi,
## dist_water = dist_water))
## Edge correction: "border"
## [border correction distance r = 0 ]
## --------------------------------------------------------------------------------
## Quadrature scheme (Berman-Turner) = data + dummy + weights
##
## Data pattern:
## Planar point pattern: 74 points
## Average intensity 7.8e-11 points per square unit
## Window: polygonal boundary
## 378 separate polygons (2 holes)
## vertices area relative.area
## polygon 1 5454 889865000000 9.38e-01
## polygon 2 19 11135700 1.17e-05
## polygon 3 6 1679950 1.77e-06
## polygon 4 59 375017000 3.95e-04
## polygon 5 25 54751800 5.77e-05
## polygon 6 5 1220870 1.29e-06
## polygon 7 8 6790020 7.16e-06
## polygon 8 63 151745000 1.60e-04
## polygon 9 12 10733900 1.13e-05
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## polygon 11 10 6152880 6.49e-06
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## polygon 13 15 12418500 1.31e-05
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## polygon 21 28 31469400 3.32e-05
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## enclosing rectangle: [273874.9, 1870573.4] x [369042.8, 1735666.4] units
## (1597000 x 1367000 units)
## Window area = 9.48262e+11 square units
## Fraction of frame area: 0.435
## Quadrature weights:
## (counting weights based on 32 x 32 array of rectangular tiles)
## All weights:
## range: [55500000, 2.13e+09] total: 9.47e+11
## Weights on data points:
## range: [59300000, 1.07e+09] total: 2.75e+10
## Weights on dummy points:
## range: [55500000, 2.13e+09] total: 9.2e+11
## --------------------------------------------------------------------------------
## FITTED :
##
## Nonstationary Poisson process
##
## ---- Intensity: ----
##
## Log intensity: ~elev + cover + hfi + dist_water
## Model depends on external covariates 'elev', 'cover', 'hfi' and 'dist_water'
## Covariates provided:
## elev: im
## cover: im
## hfi: im
## dist_water: im
##
## Fitted trend coefficients:
## (Intercept) elev cover hfi dist_water
## -2.345970e+01 -3.511481e-03 1.690825e-02 4.869412e+00 9.553740e-05
##
## Estimate S.E. CI95.lo CI95.hi Ztest
## (Intercept) -2.345970e+01 5.287726e-01 -2.449607e+01 -22.423323608 ***
## elev -3.511481e-03 4.743760e-04 -4.441241e-03 -0.002581722 ***
## cover 1.690825e-02 5.095351e-03 6.921542e-03 0.026894951 ***
## hfi 4.869412e+00 5.529194e-01 3.785710e+00 5.953114352 ***
## dist_water 9.553740e-05 5.890243e-05 -1.990924e-05 0.000210984
## Zval
## (Intercept) -44.366332
## elev -7.402317
## cover 3.318367
## hfi 8.806731
## dist_water 1.621960
##
## ----------- gory details -----
##
## Fitted regular parameters (theta):
## (Intercept) elev cover hfi dist_water
## -2.345970e+01 -3.511481e-03 1.690825e-02 4.869412e+00 9.553740e-05
##
## Fitted exp(theta):
## (Intercept) elev cover hfi dist_water
## 6.480108e-11 9.964947e-01 1.017052e+00 1.302443e+02 1.000096e+00
## Problem:
## Values of the covariate 'hfi' were NA or undefined at 0.68% (4 out of 587) of the quadrature points
## Warning: Values of the covariate 'hfi' were NA or undefined at 0.45% (12 out of
## 2667) of the quadrature points. Occurred while executing: ppm.ppp(Q =
## ppp_list[[s]], trend = ~elev + cover + hfi + dist_water,## Point process model
## Fitted to data: ppp_list[[s]]
## Fitting method: maximum likelihood (Berman-Turner approximation)
## Model was fitted using glm()
## Algorithm converged
## Call:
## ppm.formula(Q = ppp_list[[s]] ~ elev + cover + hfi + dist_water,
## covariates = list(elev = elevation, cover = cover, hfi = hfi,
## dist_water = dist_water))
## Edge correction: "border"
## [border correction distance r = 0 ]
## --------------------------------------------------------------------------------
## Quadrature scheme (Berman-Turner) = data + dummy + weights
##
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## Average intensity 8.1e-10 points per square unit
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##
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## (1597000 x 1367000 units)
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## Fraction of frame area: 0.435
## Quadrature weights:
## (counting weights based on 64 x 64 array of rectangular tiles)
## All weights:
## range: [6750000, 5.33e+08] total: 9.45e+11
## Weights on data points:
## range: [6750000, 2.92e+08] total: 6.97e+10
## Weights on dummy points:
## range: [6750000, 5.33e+08] total: 8.75e+11
## --------------------------------------------------------------------------------
## FITTED :
##
## Nonstationary Poisson process
##
## ---- Intensity: ----
##
## Log intensity: ~elev + cover + hfi + dist_water
## Model depends on external covariates 'elev', 'cover', 'hfi' and 'dist_water'
## Covariates provided:
## elev: im
## cover: im
## hfi: im
## dist_water: im
##
## Fitted trend coefficients:
## (Intercept) elev cover hfi dist_water
## -2.111907e+01 -1.478995e-03 9.975132e-03 4.838300e+00 -3.936119e-05
##
## Estimate S.E. CI95.lo CI95.hi Ztest
## (Intercept) -2.111907e+01 1.352834e-01 -2.138422e+01 -2.085392e+01 ***
## elev -1.478995e-03 9.298462e-05 -1.661242e-03 -1.296749e-03 ***
## cover 9.975132e-03 1.304182e-03 7.418982e-03 1.253128e-02 ***
## hfi 4.838300e+00 1.561870e-01 4.532179e+00 5.144421e+00 ***
## dist_water -3.936119e-05 2.095225e-05 -8.042685e-05 1.704473e-06
## Zval
## (Intercept) -156.109836
## elev -15.905807
## cover 7.648573
## hfi 30.977604
## dist_water -1.878614
##
## ----------- gory details -----
##
## Fitted regular parameters (theta):
## (Intercept) elev cover hfi dist_water
## -2.111907e+01 -1.478995e-03 9.975132e-03 4.838300e+00 -3.936119e-05
##
## Fitted exp(theta):
## (Intercept) elev cover hfi dist_water
## 6.731366e-10 9.985221e-01 1.010025e+00 1.262546e+02 9.999606e-01
## Problem:
## Values of the covariate 'hfi' were NA or undefined at 0.45% (12 out of 2667) of the quadrature points
## Warning: Values of the covariates 'elev', 'hfi' were NA or undefined at 0.4%
## (22 out of 5456) of the quadrature points. Occurred while executing: ppm.ppp(Q
## = ppp_list[[s]], trend = ~elev + cover + hfi + dist_water,## Point process model
## Fitted to data: ppp_list[[s]]
## Fitting method: maximum likelihood (Berman-Turner approximation)
## Model was fitted using glm()
## Algorithm converged
## Call:
## ppm.formula(Q = ppp_list[[s]] ~ elev + cover + hfi + dist_water,
## covariates = list(elev = elevation, cover = cover, hfi = hfi,
## dist_water = dist_water))
## Edge correction: "border"
## [border correction distance r = 0 ]
## --------------------------------------------------------------------------------
## Quadrature scheme (Berman-Turner) = data + dummy + weights
##
## Data pattern:
## Planar point pattern: 1759 points
## Average intensity 1.85e-09 points per square unit
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## enclosing rectangle: [273874.9, 1870573.4] x [369042.8, 1735666.4] units
## (1597000 x 1367000 units)
## Window area = 9.48262e+11 square units
## Fraction of frame area: 0.435
## Quadrature weights:
## (counting weights based on 90 x 90 array of rectangular tiles)
## All weights:
## range: [2760000, 2.69e+08] total: 9.45e+11
## Weights on data points:
## range: [2760000, 1.35e+08] total: 8.26e+10
## Weights on dummy points:
## range: [2760000, 2.69e+08] total: 8.63e+11
## --------------------------------------------------------------------------------
## FITTED :
##
## Nonstationary Poisson process
##
## ---- Intensity: ----
##
## Log intensity: ~elev + cover + hfi + dist_water
## Model depends on external covariates 'elev', 'cover', 'hfi' and 'dist_water'
## Covariates provided:
## elev: im
## cover: im
## hfi: im
## dist_water: im
##
## Fitted trend coefficients:
## (Intercept) elev cover hfi dist_water
## -2.046112e+01 -1.310042e-03 1.045653e-02 4.990537e+00 -3.885610e-05
##
## Estimate S.E. CI95.lo CI95.hi Ztest
## (Intercept) -2.046112e+01 9.180183e-02 -2.064105e+01 -2.028120e+01 ***
## elev -1.310042e-03 6.134897e-05 -1.430284e-03 -1.189800e-03 ***
## cover 1.045653e-02 8.686491e-04 8.754013e-03 1.215906e-02 ***
## hfi 4.990537e+00 1.087027e-01 4.777484e+00 5.203591e+00 ***
## dist_water -3.885610e-05 1.341862e-05 -6.515611e-05 -1.255610e-05 **
## Zval
## (Intercept) -222.883618
## elev -21.353940
## cover 12.037696
## hfi 45.909988
## dist_water -2.895686
##
## ----------- gory details -----
##
## Fitted regular parameters (theta):
## (Intercept) elev cover hfi dist_water
## -2.046112e+01 -1.310042e-03 1.045653e-02 4.990537e+00 -3.885610e-05
##
## Fitted exp(theta):
## (Intercept) elev cover hfi dist_water
## 1.299710e-09 9.986908e-01 1.010511e+00 1.470154e+02 9.999611e-01
## Problem:
## Values of the covariates 'elev', 'hfi' were NA or undefined at 0.4% (22 out of 5456) of the quadrature points
## Warning: Values of the covariate 'hfi' were NA or undefined at 0.67% (18 out of
## 2698) of the quadrature points. Occurred while executing: ppm.ppp(Q =
## ppp_list[[s]], trend = ~elev + cover + hfi + dist_water,## Point process model
## Fitted to data: ppp_list[[s]]
## Fitting method: maximum likelihood (Berman-Turner approximation)
## Model was fitted using glm()
## Algorithm converged
## Call:
## ppm.formula(Q = ppp_list[[s]] ~ elev + cover + hfi + dist_water,
## covariates = list(elev = elevation, cover = cover, hfi = hfi,
## dist_water = dist_water))
## Edge correction: "border"
## [border correction distance r = 0 ]
## --------------------------------------------------------------------------------
## Quadrature scheme (Berman-Turner) = data + dummy + weights
##
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## Average intensity 8.43e-10 points per square unit
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## Fraction of frame area: 0.435
## Quadrature weights:
## (counting weights based on 64 x 64 array of rectangular tiles)
## All weights:
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## Weights on data points:
## range: [5180000, 2.66e+08] total: 6.3e+10
## Weights on dummy points:
## range: [7240000, 5.33e+08] total: 8.82e+11
## --------------------------------------------------------------------------------
## FITTED :
##
## Nonstationary Poisson process
##
## ---- Intensity: ----
##
## Log intensity: ~elev + cover + hfi + dist_water
## Model depends on external covariates 'elev', 'cover', 'hfi' and 'dist_water'
## Covariates provided:
## elev: im
## cover: im
## hfi: im
## dist_water: im
##
## Fitted trend coefficients:
## (Intercept) elev cover hfi dist_water
## -2.059377e+01 -2.172077e-03 8.193666e-03 4.581051e+00 -6.675436e-05
##
## Estimate S.E. CI95.lo CI95.hi Ztest
## (Intercept) -2.059377e+01 1.402255e-01 -2.086860e+01 -2.031893e+01 ***
## elev -2.172077e-03 1.008390e-04 -2.369717e-03 -1.974436e-03 ***
## cover 8.193666e-03 1.428951e-03 5.392973e-03 1.099436e-02 ***
## hfi 4.581051e+00 1.520399e-01 4.283058e+00 4.879044e+00 ***
## dist_water -6.675436e-05 2.184644e-05 -1.095726e-04 -2.393613e-05 **
## Zval
## (Intercept) -146.861800
## elev -21.540041
## cover 5.734042
## hfi 30.130590
## dist_water -3.055617
##
## ----------- gory details -----
##
## Fitted regular parameters (theta):
## (Intercept) elev cover hfi dist_water
## -2.059377e+01 -2.172077e-03 8.193666e-03 4.581051e+00 -6.675436e-05
##
## Fitted exp(theta):
## (Intercept) elev cover hfi dist_water
## 1.138259e-09 9.978303e-01 1.008227e+00 9.761692e+01 9.999332e-01
## Problem:
## Values of the covariate 'hfi' were NA or undefined at 0.67% (18 out of 2698) of the quadrature points
For each season - assessing spatial distribution of bear occurrences (I.e., clustered, dispersed, or randomly distributed)
# Ripley's K with Envelope for each season
for (s in names(ppp_list)) {
# Compute Ripley's K for the current season
K <- Kest(ppp_list[[s]])
E <- envelope(ppp_list[[s]],
Kest,
correction = "border",
rank = 1,
nsim = 19,
fix.n = TRUE)
plot(K,
main = paste("Ripley's K -", s))
plot(E,
add = TRUE,
lty = 2)
}## Generating 19 simulations of CSR with fixed number of points ...
## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
## 19.
##
## Done.
## Generating 19 simulations of CSR with fixed number of points ...
## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
## 19.
##
## Done.
## Generating 19 simulations of CSR with fixed number of points ...
## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
## 19.
##
## Done.
## Generating 19 simulations of CSR with fixed number of points ...
## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
## 19.
##
## Done.